#420855

order	512
nilpotency class	3
isoclinism	isoclinic class of #417980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15281
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1849 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,12)(4,17)(5,22)(6,10)(7,11)(8,14)(9,19)(13,30)(15,24)(16,29)(18,23)(20,26)(21,27)(25,31)(28,32);(1,3,5,15,7,14,6,16)(2,8,10,24,11,12,22,29)(4,18,19,13,21,32,20,31)(9,25,17,28,26,30,27,23);(1,4)(2,9)(3,13)(5,20)(6,19)(7,21)(8,23)(10,27)(11,26)(12,28)(14,31)(15,18)(16,32)(17,22)(24,30)(25,29);(1,2)(3,12)(4,9)(5,22)(6,10)(7,11)(8,14)(13,28)(15,24)(16,29)(17,20)(18,30)(19,27)(21,26)(23,31)(25,32)

Polycyclic presentation:

2^9|b^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263012, #72267112
maximal normal subgroup of	#72263012, #72267112
maximal quotient of	#72262941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328051);